Deciding whether vector fields are conservative or not by computing curl F.

Question

Type in a potential function f (that is, ∇f=F). Assume the potential function has a value of zero at the origin. 

A. F(x,y)=(16x+8y)i+(8x+4y)j

f(x,y)= 

B. F(x,y)=8yi+9xj

f(x,y)= 

C. F(x,y,z)=8xi+9yj+k

f(x,y,z)= 

D. F(x,y)=(8sin⁡y)i+(16y+8xcos⁡y)j

f(x,y)= 

E. F(x,y,z)=8x2i+8y2j+2z2k

f(x,y,z)= 

Answer 1

The curl of any vector field in rectangular coordinates can be expressed in the form of a determinant. In the problem you are discussing, we take "F" to be some vector field with x, y, and z components. Assume the unit vectors are denoted as i, j, k. We can write the expression for the curl as:

curl F = ∇×F = ⌈ i j k ⌉

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det | ∂/∂x ∂/∂y ∂/∂z |

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⌊ F_x F_y F_z ⌋

You evaluate this determinant by doing a Laplace development across the first row (the unit vectors), so you wind up with a messy expression that looks like this:

curl F = ( ∂F_z/∂y - ∂F_y/∂z ) i - ( ∂F_z/∂x - ∂F_x/∂z ) j + ( ∂F_y/∂x - ∂F_x/∂y ) k

In order for a vector field to be conservative, the curl of the vector field must vanish identically everywhere. This means that each component of the curl must vanish identically. For each one of the examples you have provided, evaluate each component of the curl by using the formula above. The fields that are conservative will have zero curl.